cone

Given a topological space $X$ , the cone on $X$ (sometimes denoted by $C X$ ) is the quotient space $X\\times[0,1]/X\\times\\left\\{0\\right\\}.$ Note that there is a natural inclusion $X\\hookrightarrow CX$ which sends $x$ to $(x,1).$

If $(X,x_{0})$ is a based topological space, there is a similar reduced cone construction, given by $X\\times[0,1]/(X\\times\\left\\{0\\right\\})\\cup(\\left\\{x_{0}\\right\\}\\times[0,1]).$ With this definition, the natural inclusion $x\\mapsto(x,1)$ becomes a based map, where we take $(x_{0},0)$ to be the basepoint of the reduced cone.

单词	Cone
释义	cone Given a topological space $X$ , the cone on $X$ (sometimes denoted by $C X$ ) is the quotient space $X\\times[0,1]/X\\times\\left\\{0\\right\\}.$ Note that there is a natural inclusion $X\\hookrightarrow CX$ which sends $x$ to $(x,1).$ If $(X,x_{0})$ is a based topological space, there is a similar reduced cone construction, given by $X\\times[0,1]/(X\\times\\left\\{0\\right\\})\\cup(\\left\\{x_{0}\\right\\}\\times[0,1]).$ With this definition, the natural inclusion $x\\mapsto(x,1)$ becomes a based map, where we take $(x_{0},0)$ to be the basepoint of the reduced cone.
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